Write the equation for a parabola with a focus at $(-3,-5)$ and a directrix at $x=-7$. $x=$
Explanation: The strategy A parabola is defined as the set of all points that are the same distance away from a point (the focus) and a line (the directrix). Let $(x,y)$ be a point on the parabola. Then the distance between $(x,y)$ and the focus, $(-3,-5)$, is equal to the distance between $(x,y)$ and the directrix, $x=-7$. Once we find these distances, we can equate them in order to derive the equation of our parabola. Finding the distances from $(x,y)$ to the focus and the directrix The distance between $(x,y)$ and $(-3,-5)$ is $\sqrt{(x+3)^2+(y+5)^2}$. [How did we find that?] Similarly, the distance between $(x,y)$ and the line $x=-7$ is $\sqrt{(x+7)^2}$. [How did we know that?] Deriving the formula by equating the distances $\begin{aligned} \sqrt{(x+7)^2} &= \sqrt{(x+3)^2+(y+5)^2} \\\\ (x+7)^2 &= (x+3)^2+(y+5)^2 \\\\ {x^2}+14x{+49} &= {x^2}{+6x}+9+(y+5)^2\\\\ 14x{-6x}&=(y+5)^2+9{-49} \\\\ 8x&=(y+5)^2-40\\\\ x&=\dfrac{(y+5)^2}{8}-5\end{aligned}$ The answer The equation of our parabola is $x=\dfrac{(y+5)^2}{8}-5$. Here is the graph of our parabola. As expected, the distance between a point on the parabola, $(x,y)$, and the focus is the same as the distance between $(x,y)$ and the directrix. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${\llap{-}10}$ ${\llap{-}11}$ ${\llap{-}12}$ ${\llap{-}13}$ ${\llap{-}14}$ $y$ $x$ ${(x,y)}$